Q. How do you proceed in the operation?

A. Multiply the second and third terms together, and divide their product by the first term; the quotient will be the fourth term, or answer, in the same denomination with the third term.

Q. How may this process of multiplying and dividing be, in most cases, materially shortened?

A. By multiplying the third term by the ratio of the first and second, expressed either as a fraction in its lowest terms, or as a whole number.

Q. If the result, or fourth term, be not in the denomination required, what is to be done?

A. It may be brought to it by Reduction.

Q. If there be a remainder in dividing by the first term, or mult§plying by the ratio, what is to be done with it?

A. Reduce it to the next lower denomination, and divide again, and so on, till it can be reduced no more.*

*

*As this rule is commonly divided into direct and inverse, it may not be amiss, for the benefit of some teachers, to explain how they may be distinguished; also, to give the rule for each."

The Rule of Three Direct is when more requires more, or less requires less. It may be known thus: more requires more, when the third term is more than the first, and requires the fourth term, or answer, to be more than the second and less requires less, when the third term is less than the first, and requires the fourth term, or answer, to be less than the second.

RULE 1. State the question, that is, place the numbers so that the first and third terms may be of the same name, and the second term of the same name with the answer, or thing sought.

2. Bring the first and third terms to the same denomination, and reduce the second term to the lowest denomination mentioned in it.

3. Divide the product of the second and third terms by the first term; the quotient will be the answer to the question, in the same denomination with the second term, which may be brought into any other denomination required. The Rule of Three Inverse is, when more requires less, or less requires more, and may be known thus: more requires less, when the third term is more than the first, and requires the fourth term, or answer, to be less than the second; and less requires more, when the third term is less than the first, and requires the fourth term to be more than the second.

RULE. State and reduce the terms as in the Rule of Three Direct; then multiply the first and second terms together, and divide their product by the third term; the quotient will be the answer, in the same denomination with the middle term.

Note.-Although the distinction of direct and inverse is frequently made, still it is totally useless. Besides, this mode of arranging the proportional numbers is very erroneous, and evidently calculated to conceal from the view of the pupil the true principles of ratio, and, consequently, of proportion, on which the solution proceeds. The following example will render the absurdity more apparent>

A certain rich farmer gave 20 sheep for a sideboard; how many sideboards may be bought for 100 sheep?

20 sheep: 1 sideboard: 100 sheep: 5 sideboards, the 4th term, or Ans

More Exercises for the Slate.

25. If 600 bushels of wheat cost $1200, what will 3600 bush els cost? and what is the ratio of the 1st and 2d terms?

Perform the foregoing example, and the following, first, with out finding the ratio; then, by finding the ratio, and multiply ing by it.

A. $7200. The ratio, 36026Xx 1200 $7200, the same. 26. How many bushels of wheat may be bought for $7200, if 600 bushels cost $1200? A. 3600 bushels. Ratio, 6; then, 6 x 600 3600 bushels.

27. If $7200 buy 3600 bushels of wheat, what will 600 bushels cost? A. $1200. Ratio, .

28. If board for 1 year, or 52 weeks, amount to $182, what will 39 weeks come to? A. $136,50. Ratio, & × 182=$136,50, the same.

29. If 30 bushels of rye may be bought for 120 bushels of potatoes, how many bushels of rye may be bought for 600 bushels of potatoes? A. 150 bushels rye. Ratio, 5.

30. If 4 cwt. 1 qr. of sugar cost $45,20, what will 21 cwt. 1 gr. cost? (Bring 4 cwt. I qr., and 21 cwt. 1 qr. into quarters first.) A. $226. Ratio, 5.

31. If I buy 60 yards of cloth for $120, what is the cost per yard?-2. What is the cost per ell Flemish ?-150. What per ell English?-250. What per ell French ?-3. A. $9.

32. Bought 4 tuns of wine for $322,56; what did 1 pipe cost?4032. What did 1 hhd. ?-2016. What did 1 tierce?-1344. What did 1 bbl.?-1008. A. $84. What did 2 quarts cost 16. What did 3 pints cost?-12. What did 4 gills cost?-4.

A. $,32.

33. Bought 6 tuns of wine for $500,50; what did 1 pipe cost: A. $41,708+.

34. When a merchant compounds with his creditors for 40 cents on the dollar, how much is A's part, to whom he owes $2500? How much is B's part, to whom he owes $1600?

A. A's, $1000; B's, $640.

35. A, failing in trade, owes the following sums, viz. to B, $1600,60; to C, $500; to D, $750,20; to E, $1000; to F, $230; and his property, which is worth no more than $1020,20, he gives up to nis creditors; how much does he pay on the dollar? and what is the amount of loss sustained by all?

A. 25 cents on $1; and the amount of loss is $3060,60.

It must appear evident, to every rational mind, that there can be no analogy between 20 sheap and 1 sideboard, or 100 sheep and 5 sideboaras. With the same propriety it may be asked, what ratio or analogy there is between such heterogeneous quantities as 2 monkeys and 5 merino shawls; or between 7 lob stors and 4 bars of music; the one is equally as correct as the other

36. Bought 4 tierces of rice, each weighing 7 cwt. 2 qrs. 16 Ibs.; what do they come to at $9,35 per cwt. ?-285842. At $2,50 per qr.?-305714. At 10 cents per pound?-34240.

A. $933,956+.

37. Bought, by estimation, 300 yards of cloth, for $450,60; but, by actual measurement, there were no more than 275 yds 2 qrs.; for how much must I sell the measured yards per yd., so as to neither make nor lose ?-1635. How much must I self 20 yds. for, so as to lose nothing?-32711. How much 25 yds. 2 qrs. ?-41707. How much 30 yds. 1 qr. 3 na. ?-49782.

A. $125,835+. 38. If a staff 6 feet long cast a shade on level ground 9 feet, what is the height of that steeple whose shade measures at the same time 198 feet? A. 132 feet in height. Ratio, 22.

39. If 3 cwt. 2 qrs. 16 lbs. of sugar cost $51, what will 10 cwt. 3 qrs. 20 lbs. cost?-Ratio, 3 X 51=153. What will 21 cwt. 3 qrs. 12 lbs. ?-6, ratio, X 51306. What will 43 cwt. 2 qrs. 24 lbs. ?-Ratio, 12; then, 12 × 51=612. A. $1071.

Note. The following examples may be performed either by analysis, or by finding the ratio, or by the common rule. Perhaps it would be well to let the pupil take his choice. The one by ratio is recommended.

40. If $100 gain $6 in a year, how much will $20 gain in the same time?-120. Ratio, . How much will $10 gain?60. Ratio, How much will $50 gain?-3. Ratio, . How much will $75 gain?-450. Ratio,. What will $200 gain12. Ratio, 2. How much will $300 gain?-18. Ratio, 3. How much will $500 gain?-30. Ratio, 5. How much will $800 gain?-48. Ratio, 8. How much will $1000 gain?-60. Ratio, 10. How much will $1250 gain ?-75. Ratio, 12. How much will $2000 gain ?-120. Ratio, 20. A. $372,30.

41. If 12 men can build a wall in 20 days, how many can do the same in 5 days? A. 48 men. Ratio, 4.

42. If 60 men can build a wall in days, how many men can do the same in 20 days? A. 12 men. Ratio, .

43. If 4 men can build a wall in 120 days, in how many days will 12 men do the same ?-40. Will 16 men?-30. Will 20 men?-24. Will 24 men ?-20. A. 114 days.

44. If a man perform a journey in 30 days, by travelling 6 hours each day, in how many days will he perform it by travelling 10 hours each day? (10 hours will require a less number of days than 6 hours; that is, the multiplying or 2d term must be the smallest.) A. 18 days. Ratio, x 30 18, the same. 45. If a field will keep 2 cows 20 days, how long will it keep 5 cows?-8. Will it keep 8 cows?-5. Will it keep 10 cows?4. Will it keep 20 cows?-2. A. 19 days.

=

46. If 60 bushels of grain, at $1 per bushel, will pay a debt, how many bushels, at $1,50 a bushel, will pay the same ?-40.

How many bushels at $1,202-50. How many at 80 cents?-75. At 50 cents?-120. At 40 cents?-150. At 30 cents?-200.

A. 635 bushels.

47. How much in length that is 6 inches in breadth will take a square foot? (12 inches in length and 12 in breadth make 1 square foot; then, 6 inches in breadth will require more in length; that is, 6: 12 12.)--24. How many 4 inches in breadth?-36. How many 8 inches in breadth ?-18. How many

16 inches in breadth ?-9. A. 87 inches.

48. If a man's income be $1750 a year, how much may he spend each day to lay up $400 a year? A. $3,70.

49. If 6 shillings make $1, New England currency, how much will 4 s. 6 d. make, in federal money ?-,75. Will 2s. 6d. ?,41. Will 1 s. 6d. ?-,25. Will 3 s. 9 d.?-,62. A. $2,041.

50. A merchant bought 26 pipes of wine on 6 months' credit, but, by paying ready money, he got it 3 cents a gallon cheaper; how much did he save by paying ready money?-A. $98,28.

51. Bought 400 yards 2 qrs. of plaid for $406,80, but could sell it for no more than $300; what was my loss per ell French? A. $,40.

52. If 120 gallons of water, in 1 hour, fall into a cistern con taining 600 gallons, from which, by 1 pipe, 20 gallons run out in 1 hour, and by another 50 gallons, in what time will the cistern be filled? A. 12 hours.

53. A merchant bought 40 pieces of broadcloth, each piece containing 45 yards, at the rate of $6 for 9 yards, and sold it again at the rate of $15 for 18 yards; how much did he make in trading? A. $300.

54. A borrowed of B $600 for 3 years; how long ought A to lend B $800 to requite the favor?-2-3. How long ought he to lend him $900 ?-2. How long $500?-3-7-6. How long $1200?1-6. A. 9 years, 4 mo. 6 days.

55. A gentleman bought 3 yards of broadcloth 14 yards wide; how many yards of flannel, which is only yd. wide, will line

the same?

It is evident it will take more cloth which is only yd. wide, than if it were 14 yd. wide; hence 14 must be the middle term. A. 6 yds. Ratio, 2.

56. A regiment of soldiers, consisting of 800 men, are to be clothed, each suit containing 43 yds. of cloth, which is 13 yd. wide, and lined with flannel yd. wide; how many yards of flannel will be sufficient to line all the suits?

A. 8633 yds. 1 qr. 1 nå.

FRACTIONS. 57. If of a barrel of flour cost of a dollar, what will of a barrel cost?

Banalysis. It is plain that, if we knew the price of 1 barrel, of a barrel would cost as much. If of a barrel cost of a dollar, §, or 1 barrel, will cost 8 times as much, that is, 40 X 3 $13, Ans.

8 X 5

16

16 X 4

Or, as is more than, we may make the 2d, or multiply Ing term, as in the foregoing examples, thus:

Bbls. Bbls. $

(Inverting by

Or, multiplying by the ratio, thus; the ratio of to is ÷

246, ratio; then,

6 x 5 30

16 16

$13, Ans. as before.

Or, which is obviously the same, having inverted the 1st, or dividing term, multiply all the fractions together; that is, proceed as in Division of Fractions, (¶XLVII.) thus,

8 x 3 x 5 120

1 X 4 X 16

64

$13, Ans., as before.

The pupil may perform the following examples by either of the preceding methods, but the one by analysis is recommended, it being the best exercise for the mind.

58. If 3 lbs. of butter cost § of a dollar, what cost lb.? A. $16. 59. If of a bushel of wheat cost

bushel cost? A. $16.

of a dollar, what will 1

60. If 1 yds. of cloth cost $12, what will 1 yd. cost? A. $. 61. At a pound, what will 40 pounds cost? A. $24. 62. If yd. cost $2, what will 1 yd, cost? A. $2,82}. 63. If of 3 yd. cost $2, what is it a yard? A. $5.

64. If of of 1 of $1 buy 20 apples, how many apples will $5 buy? A. 487 apples.

65. If oz. of gold be worth $1,50, what is the cost of 1 oz.?

A. $1,80.

66. If 163 yds. will make 8 coats, how many yards will it take for 1 coat? A. 274 yds.

67. If of of a gallon cost $, what will 5

gallons cost?

A. $9.

A. $133.

68. If 6 yds. cost $53, what will 14 yds. cost? 69. If of cwt. of sugar cost $ro, what will 40 ewt. cost?

A. $824.

70. If yd. of silk cost of $, what is the price of 50 yds.?

17

4 $31.